Problem: When the two-digit integer $MM$, with equal digits, is multiplied by the one-digit integer $M$, the result is the three-digit integer $NPM$. What is the greatest possible value of $NPM$?
Solution: $M=1$, $5$, or $6$ since no other digits have the property that the units digit of $M\times M$ is $M$. Therefore, the greatest possible value of $MM\times M=NPM$ is $66\times6=\boxed{396}$.